rays pq and qr are perpendicular. point s lies in the interior of <pqr. if m<pqs = 4x + 7 and m<sqr = 9 + 4a, find m<pqs and m<sqr
To find the measures of angles <PQS and <SQR, we need to use the given equations:
1. The measure of <PQS is represented as 4x + 7.
2. The measure of <SQR is represented as 9 + 4a.
The information also states that rays PQ and QR are perpendicular, and point S lies in the interior of <PQR. This means that <PQS and <SQR are adjacent angles, and their sum should be equal to 90 degrees since they are complementary.
So, we can write the equation:
4x + 7 + 9 + 4a = 90
Simplify the equation:
4x + 4a + 16 = 90
Next, combine like terms:
4x + 4a = 74
Divide both sides of the equation by 4:
x + a = 18.5
Now, the equation 4x + 7 represents <PQS. To find the measure of <PQS, substitute the value of x back into the equation:
4(18.5) + 7 = 74 + 7 = 81
So, m<PQS = 81 degrees.
Similarly, the equation 9 + 4a represents <SQR. To find the measure of <SQR, substitute the value of a back into the equation:
9 + 4(18.5) = 9 + 74 = 83
So, m<SQR = 83 degrees.
Therefore, <PQS measures 81 degrees and <SQR measures 83 degrees.